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Ancient numeral systems
This page lists the various numeral systems used in ancient times. Egyptian numerals The Egyptian numerals were invented in ancient Egypt about 3000 BC. Egyptian numerals system uses additive principle, for example: = 100+100+10+10+1+1+1 = 223 The glyphs for addition and subtraction are and , respectively. Greek numerals Attic numerals The Attic numerals appeared in the 7th century BC. Ionic numerals In the 4th century BC the attic numerals were replaced by the ionic numerals. Greek numerals system uses additive principle, for example: ͵θτπεʹ = 9000+300+80+5 = 9,385 Ten thousand (one myriad) ancient greeks wrote as "M" and quantity of myriads they wrote before "M". For example: ͵θτπεM͵θτπεʹ = 93,859,385 Thus, the largest number which ancient greeks could write was ,θϡϟθM,θϡϟθ' = 99,999,999. Notation of Archimedes Earliest googological notation in human history was invented by Archimedes (c. 287 – c. 212 BC) with aim to calculate how much grains of sand the Universe can contain (in supposition, that all universe filled by sand). Definition All natural numbers up to myriada of myriads (100,000,000) are first numbers (ἀριθμοὶ ἐς τὰς μυρίας μυριάδας πρώτοι καλουμένοι). The unit of (n+1)ths numbers is myriada of myriads of n-ths numbers. And so on up to 100,000,000-ths numbers. Examples *ιʹ μονάδες τῶν δευτέρων ἀριθμῶν = ten of units of second numbers \(=10\times 10^{8\times(2-1)}= 10^{9}\) *μυρίαι μυριάδες τῶν δευτέρων ἀριθμῶν = myriad of myriads of second numbers=αʹ μονάδα τῶν τρίτων ἀριθμῶν = one unit of third order \(=10^{8\times(3-1)}= 10^{16}\) *ιʹ μυριάδες τῶν τρίτων ἀριθμῶν = ten myriads of third numbers \(=10\times10^4\times10^{8\times(3-1)}= 10^{21}\) *ιʹ μονάδες τῶν πέμπτων ἀριθμῶν = ten of units of fifth numbers \(=10\times 10^{8\times(5-1)}= 10^{33}\) *ιʹ μυριάδες τῶν ἕκτων ἀριθμῶν = ten myriads of sixth numbers \(=10\times10^4\times10^{8\times(6-1)}=10^{45}\) *͵α μονάδες τῶν ἑβδόμων ἀριθμῶν = thousand of units of seventh numbers \(=10^3\times10^{8\times(7-1)}=10^{51}\) = χιλίαι μονάδες τῶν ἑβδόμων ἀριθμῶν *͵α μυριάδες τῶν ὀγδόων ἀριθμῶν = thousand of myriads of eighth numbers= \(=10^3\times10^4\times10^{8\times(8-1)}=10^{63}\) *͵θτπεʹ μονάδες τῶν μυριακισμυριοστῶν ἀριθμῶν = 9,385 units of 100,000,000-ths numbers = \(9,385 \times 10^{799,999,992}\) In general, each natural number up to \(10^{800,000,000}\) can be uniquely written in this form: \(N=\sum_{i=1}^k\)(Number of myriads of i-ths numbers+Number of units of i-ths numbers) Extension All natural numbers up to myriada of myriads of 100,000,000ths numbers are numbers of first period (ἀριθμοὶ πρώτας περιόδου). The unit of first numbers of (n+1)th period is myriada of myriads of 100,000,000ths numbers of the n-th period. μυρίαι μυριάδες τᾶς δευτέρας περιόδου πρώτων ἀριθμῶν μονὰς καλείσθω τᾶς δευτέρας περιόδου δευτέρων ἀριθμῶν = the myriad of myriads of first numbers of 2nd period is equal to the unit of 2-ths numbers of 2nd period \(=10^{8\times 10^8 +8}\) \(10^{16\times 10^8}\) is unit of first numbers of 3rd period \(10^{24\times 10^8}\) is unit of first numbers of 4th period \(10^{32\times 10^8}\) is unit of first numbers of 5th period and so on, up to: μυριακισμυριοστᾶς περιόδου μυριακισμυριοστῶν ἀριθμῶν μυρίας μυριάδας = myriad of myriads of 100,000,000th numbers of 100,000,000th period \(=10^{8\times 10^{16}}\) Roman numerals Roman numerals system uses additive principle, for example: MMMCCCXXXIII=MMM+CCC+XXX+III=3000+300+30+3=3,333. Subtractive notation is also used, in the following cases: I placed before V or X reduces number by one, X placed before L or C reduces number by ten, C placed before D or M reduces number by hundred, The largest number representible in Roman numerals is: MMMMCMXCIX = 4,000+(1,000-100)+(100-10)+(10-1) = 4,000+900+90+9 = 4,999. Chinese numerals Chinese numerals are numerals of ancient and modern usage in China. The invention of chinese numerals and math calculations traditionally ascribed to the mythological Yellow Emperor (2698–2598 BC). The numbers of everyday usage Multiplicative principle Chinese numerals system uses a multiplicative principle, for example: *一千一百五十八 = 1,158 *八 = 8 *八十八 = 88 *八百八十八 = 888 *八千八百八十八 = 8,888 *八千〇八十 = 8,080 Notations for large numbers Notation 1: *\(a_0=10^4\) *\(a_{n+1}=a_n\cdot10\) Notation 2: *\(a_0=10^4\) *\(a_{n+1}=a_n\cdot10^4\) Notation 3: *\(a_0=10^8\) *\(a_{n+1}=a_n\cdot10^8\) Notation 4: *\(a_0=10^4\) *\(a_{n+1}=a_n^2\) Names and values of the large numbers Avatamsaka Sutra's notation In Avatamsaka Sutra, written in the India before 5 century AD, Buddha gave the name "Untold" for number which is defined as follows: *\(a_0=10^{10}\) *\(a_{n+1}=a_n^2\) *Untold \(=a_{122}\), or in other words, Untold \(=10^{10\times2^{122}}\). Sources Category:Notations Category:Numbers Category:Lists